Classical irregular blocks, Hill's equation and PT-symmetric periodic complex potentials

The Schrödinger eigenvalue problems for the Whittaker-Hill potential Q₂(x)=(1/2)h²cos 4x+4hμcos 2x and the periodic complex potential Q₁(x)=(1/4)h²e^−4ix+2h²cos 2x are studied using their realizations in two-dimensional conformal field theory (2dCFT). It is shown that for the weak coupling (small) h∈ℝ and non-integer Floquet parameter ν∉ℤ spectra of hamiltonians H_i = −d²/dx²+Q_i(x), i=1,2 and corresponding two linearly independent eigenfunctions are given by the classical limit of the "single flavor" and "two flavors" (N_f=1,2) irregular conformal blocks. It is known that complex non-hermitian hamiltonians which are PT-symmetric (= invariant under simultaneous parity P and time reversal T transformations) can have real eigenvalues. The hamiltonian H₁ is PT-symmetric for h,x∈ℝ. It is found that H₁ has a real spectrum in the weak coupling region for ν∈ℝ∖ℤ. This fact in an elementary way follows from a definition of the N_f=1 classical irregular block. Thus, H₁ can serve as yet another new model for testing postulates of PT-symmetric quantum mechanics.